# Tail probabilities and partial moments for quadratic forms in multivariate generalized hyperbolic random vectors

*Simon Broda*

No 13-04, UvA-Econometrics Working Papers from Universiteit van Amsterdam, Dept. of Econometrics

**Abstract:**
Countless test statistics can be written as quadratic forms in certain random vectors, or ratios thereof. Consequently, their distribution has received considerable attention in the literature. Except for a few special cases, no closed-form expression for the cdf exists, and one resorts to numerical methods. Traditionally the problem is analyzed under the assumption of joint Gaussianity; the algorithm that is usually employed is that of Imhof (1961). The present manuscript generalizes this result to the case of multivariate generalized hyperbolic (MGHyp) random vectors. The MGHyp is a very flexible distribution which nests, among others, the multivariate t, Laplace, and variance gamma distributions. An expression for the first partial moment is also obtained, which plays a vital role in financial risk management. The proof involves a generalization of the classic inversion formula due to Gil-Pelaez (1951). Two applications are considered: first, the finite-sample distribution of the 2SLS estimator of a structural parameter. Second, the Value at Risk and Expected Shortfall of a quadratic portfolio with heavy-tailed risk factors.

**Date:** 2013-05-01

**New Economics Papers:** this item is included in nep-ecm and nep-rmg

**References:** Add references at CitEc

**Citations:** View citations in EconPapers (5) Track citations by RSS feed

**Downloads:** (external link)

http://ase.uva.nl/binaries/content/assets/subsites ... ics/dp-2013/1304.pdf (application/pdf)

**Related works:**

Working Paper: Tail Probabilities and Partial Moments for Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors (2013)

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:ame:wpaper:1304

Access Statistics for this paper

More papers in UvA-Econometrics Working Papers from Universiteit van Amsterdam, Dept. of Econometrics Dept. of Econometrics, Universiteit van Amsterdam, Valckenierstraat 65, NL - 1018 XE Amsterdam, The Netherlands. Contact information at EDIRC.

Bibliographic data for series maintained by Noud P.A. van Giersbergen ().